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## Proofs: Parallelograms

# Proof: Opposite angles of a parallelogram

CCSS.Math:

## Video transcript

What I want to do
in this video is prove that the opposite
angles of a parallelogram are congruent. So for example, we want to prove
that CAB is congruent to BDC, so that that angle is
equal to that angle, and that ABD, which
is this angle, is congruent to DCA, which
is this angle over here. And to do that, we
just have to realize that we have some
parallel lines, and we have some transversals. And the parallel lines
and the transversals actually switch roles. So let's just continue these
so it looks a little bit more like transversals
intersecting parallel lines. And really, you could
just pause it for yourself and try to prove it, because
it really just comes out of alternate interior angles
and corresponding angles of transversals
intersecting parallel lines. So let's say that this
angle right over here-- let me do it in a new
color since I've already used that yellow. So let's start right
here with angle BDC. And I'm just going
to mark this up here. Angle BDC, right over here-- it
is an alternate interior angle with this angle right over here. And actually, we could
extend this point over here. I could call this
point E, if I want. So I could say angle CDB
is congruent to angle EBD by alternate
interior angles. This is a transversal. These two lines are parallel. AB or AE is parallel to CD. Fair enough. Now, if we kind of change
our thinking a little bit and instead, we now view BD
and AC as the parallel lines and now view AB as
the transversal, then we see that
angle EBD is going to be congruent to
angle BAC, because they are corresponding angles. So angle EBD is going to
be congruent to angle BAC, or I could say CAB. They are corresponding angles. And so if this angle is
congruent to that angle and that angle is
congruent to that angle, then they are congruent
to each other. So angle-- let me make sure
I get this right-- CDB, or we could say BDC, is
congruent to angle CAB. So we've proven this first
part right over here. And then to prove that
these two are congruent, we use the exact same logic. So first of all, we view
this as a transversal. We view AC as a
transversal of AB and CD. And let me go here and let
me create another point here. Let me call this point
F right over here. So we know that
angle ACD is going to be congruent to
angle FAC because they are alternate interior angles. And then we change our
thinking a little bit. And we view AC and BD
as the parallel lines and AB as a transversal. And then angle FAC is going
to be congruent to angle ABD, because they're
corresponding angles. Angle F to angle ABD, and
they are corresponding angles. So in the first time, we viewed
this as the transversal, AC as a transversal of AB and
CD, which are parallel lines. Now AB is the transversal and BD
and AC are the parallel lines. And obviously, if this
is congruent to that, and that is congruent
to that, then these two have to be congruent
to each other. So we see that if we have
opposite angles are congruent-- or if we have a
parallelogram, then the opposite angles are
going to be congruent.